Truncation Dimension for Function Approximation
We consider the approximation of functions of s variables, where s is very large or infinite, that belong to weighted anchored spaces. We study when such functions can be approximated by algorithms designed for functions with only very small number dimtrnc(ε, s) of variables. Here ε is the error demand and we refer to dimtrnc(ε, s) as the ε-truncation dimension. We show that for sufficiently fast decaying product weights and modest error demand (up to about ε ≈ 10−5) the truncation dimension is surprisingly very small.
The authors would like to thank two anonymous referees for their remarks that helped improving the presentation of the results in the paper.
P. Kritzer is supported by the Austrian Science Fund (FWF) Project F5506-N26 and F. Pillichshammer by the Austrian Science Fund (FWF) Project F5509-N26. Both projects are parts of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.